Can no longer import .MOV files?

Me will accept .mov files. When exporting from Motion, make sure to set the video codec in settings to H264. You should now hove no trouble importing and converting the .mov file to mp4 (or whatever format you choose) in Me.

One issue may be needing to add ".mov" to the filename if it is not already present.  Of course, if that's already in the filename, then that's not the issue.

Also, Apple deprecated QuickTime as a system driver for time-based media back in 2013, leaving it up to the host software to support the CODEC being used in the movie file.  If this is what's causing it, you'll need to verify which CODEC is being used in your movie files.  Here's a list of what's supported in Media Encoder, Premiere Pro and After Effects: Dropped support for QuickTime 7 era formats and codecs

If the CODEC you're using isn't listed on that page, you'll have to convert your files.  If running macOS X, the easiest way to do this is by opening the file in QuickTime Player X, allowing it to be converted and then saving the new file.  If you have lot of files, this is a bit arduous and it may be worth purchasing Apple Compressor to automate the conversion.  It may be important to note that QuickTime Player X doesn't allow the user to pick the CODEC for the converted file.  It's determined by QuickTime Player X and is either H264 for Apple ProRes.  Another option is to install Adobe Media Encoder CC2017 or earlier and use that to convert to H.264, Apple ProRes (this is going to be Mac only with CC2017), Animation, or GoPro Cineform (any CODEC that is "supported for import" should work - but some are better for edtiing).

However, even if a CODEC is listed as suppored by Premiere Pro, After Effects and Media Encoder, it may not decompress/open as expected.  I've run into this with DV files that were captured a very, very long time ago.  The solution was to use QuickTime Player X to convert the legacy DV-NTSC files and then use the converted files in Premiere Pro, After Effects or Media Encoder.

-Warren